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Solution Manual for Precalculus, 6th Edition

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    S OLUTIONS M ANUAL B EVERLY F USFIELD C OLLEGE A LGEBRA & T RIGONOMETRY S IXTH E DITION and P RECALCULUS S IXTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans David I. Schneider University of Maryland Callie J. Daniels St. Charles Community College

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    CONTENTS R Review Of Basic Concepts R.1 Sets ............................................................................................................................. 1 R.2 Real Numbers and Their Properties ........................................................................... 5 R.3 Polynomials .............................................................................................................. 12 R.4 Factoring Polynomials.............................................................................................. 19 R.5 Rational Expressions ................................................................................................ 27 R.6 Rational Exponents ..................................................................................................34 R.7 Radical Expressions .................................................................................................41 Chapter R Review Exercises ........................................................................................... 47 Chapter R Test ................................................................................................................ 52 1 Equations And Inequalities 1.1 Linear Equations ....................................................................................................... 55 1.2 Applications and Modeling with Linear Equations ..................................................59 1.3 Complex Numbers .................................................................................................... 65 1.4 Quadratic Equations .................................................................................................. 71 Chapter 1 Quiz (Sections 1.1 − 1.4).................................................................................. 82 1.5 Applications and Modeling with Quadratic Equations .............................................83 1.6 Other Types of Equations and Applications ............................................................. 92 Summary Exercises on Solving Equations ................................................................... 113 1.7 Inequalities ..............................................................................................................117 1.8 Absolute Value Equations and Inequalities ............................................................140 Chapter 1 Review Exercises ......................................................................................... 147 Chapter 1 Test ............................................................................................................... 165 2 Graphs and Functions 2.1 Rectangular Coordinates and Graphs...................................................................... 170 2.2 Circles ..................................................................................................................... 179 2.3 Functions ................................................................................................................. 187 2.4 Linear Functions ..................................................................................................... 195 Chapter 2 Quiz (Sections 2.1 − 2.4)................................................................................ 203 2.5 Equations of Lines and Linear Models ................................................................... 204 Summary Exercises on Graphs, Circles, Functions, and Equations ............................. 213 2.6 Graphs of Basic Functions ...................................................................................... 217 2.7 Graphing Techniques .............................................................................................. 223 Chapter 2 Quiz (Sections 2.5 − 2.7)................................................................................ 237 2.8 Function Operations and Composition ...................................................................238 Chapter 2 Review Exercises ......................................................................................... 249 Chapter 2 Test ............................................................................................................... 260

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    3 Polynomial and R ational Functions 3.1 Quadratic Function and Models ..............................................................................264 3.2 Synthetic Division...................................................................................................285 3.3 Zeros of Polynomial Functions ...............................................................................292 3.4 Polynomial Functions: Graphs, Applications, and Models ....................................313 Summary Exercises on Polynomial Functions, Zeros, and Graphs ..............................336 3.5 Rational Functions: Graphs, Applications, and Models .........................................350 Chapter 3 Quiz (Sections 3.1 − 3.5)................................................................................374 Summary Exercises on Solving Equations and Inequalities .........................................378 3.6 Variation .................................................................................................................384 Chapter 3 Review Exercises .........................................................................................390 Chapter 3 Test ...............................................................................................................410 4 Inverse, Exponential, and Logarithmic Functions 4.1 Inverse Functions ....................................................................................................417 4.2 Exponential Functions ............................................................................................432 4.3 Logarithmic Functions ............................................................................................449 Summary Exercises on Inverse, Exponential, and Logarithmic Functions ..................460 4.4 Evaluating Logarithms and the Change-of-Base Theorem .....................................463 Chapter 4 Quiz (Sections 4.1 − 4.4)................................................................................470 4.5 Exponential and Logarithmic Equations .................................................................471 4.6 Applications and Models of Exponential Growth and Decay ................................485 Summary Exercises on Functions: Domains and Defining Equations .........................493 Chapter 4 Review Exercises .........................................................................................497 Chapter 4 Test ...............................................................................................................504 5 Trigonometric Functions 5.1 Angles .....................................................................................................................507 5.2 Trigonometric Functions.........................................................................................514 5.3 Trigonometric Function Values and Angle Measures ............................................530 Chapter 5 Quiz (Sections 5.1–5.3) ................................................................................545 5.4 Solutions and Applications of Right Triangles .......................................................546 Chapter 5 Review Exercises .........................................................................................564 Chapter 5 Test ...............................................................................................................572 6 The Circular Functions and Their Graphs 6.1 Radian Measure ......................................................................................................577 6.2 The Unit Circle and Circular Functions ..................................................................586 6.3 Graphs of the Sine and Cosine Functions ...............................................................598 6.4 Translations of the Graphs of the Sine and Cosine Functions ................................608 Chapter 6 Quiz (Sections 6.1 − 6.4)................................................................................622 6.5 Graphs of the Tangent and Cotangent Functions ....................................................625 6.6: Graphs of the Secant and Cosecant Functions .......................................................636 Summary Exercises on Graphing Circular Functions ...................................................644

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    6.7 Harmonic Motion .................................................................................................... 647 Chapter 6 Review Exercises ......................................................................................... 653 Chapter 6 Test ............................................................................................................... 666 7 Trigonometric Identities 7.1 Fundamental Identities ............................................................................................672 7.2 Verifying Trigonometric Identities ......................................................................... 680 7.3 Sum and Difference Identities ............................................................................... 691 Chapter 7 Quiz (Sections 7.1 − 7.3)................................................................................ 705 7.4 Double-Angle and Half-Angle Identities ................................................................ 707 Summary Exercises on Verifying Trigonometric Identities ......................................... 719 7.5 Inverse Circular Functions ...................................................................................... 724 7.6 Trigonometric Equations ........................................................................................ 736 Chapter 7 Quiz (Sections 7.5–7.6) ................................................................................ 754 7.7 Equations Involving Inverse Trigonometric Functions .......................................... 756 Chapter 7 Review Exercises ......................................................................................... 763 Chapter 7 Test ............................................................................................................... 782 8 Applications of Trigonometry 8.1 The Law of Sines .................................................................................................... 788 8.2 The Law of Cosines ................................................................................................ 801 Chapter 8 Quiz (Sections 8.1 − 8.2)................................................................................ 814 8.3 Geometrically Defined Vectors and Applications ..................................................815 8.4 Algebraically Defined Vectors and the Dot Product .............................................. 823 Summary Exercises on Applications of Trigonometry and Vectors ............................ 828 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients.........830 8.6 DeMoivre’s Theorem; Powers and Roots of Complex Numbers ........................... 840 Chapter 8 Quiz (Sections 8.3 − 8.6)................................................................................ 853 8.7 Polar Equations and Graphs ....................................................................................854 8.8 Parametric Equations, Graphs, and Applications ................................................... 868 Chapter 8 Review Exercises ......................................................................................... 877 Chapter 8 Test ............................................................................................................... 891 9 Systems and Matrices 9.1 Systems of Linear Equations .................................................................................. 897 9.2 Matrix Solution of Linear Systems ......................................................................... 925 9.3 Determinant Solution of Linear Systems ................................................................ 946 9.4 Partial Fractions ...................................................................................................... 970 Chapter 9 Quiz (Sections 9.1 − 9.4)................................................................................ 985 9.5 Nonlinear Systems of Equations ............................................................................. 989 Summary Exercises on Systems of Equations ............................................................1005 9.6 Systems of Inequalities and Linear Programming ................................................1013 9.7 Properties of Matrices ........................................................................................... 1032 9.8 Matrix Inverses ..................................................................................................... 1045 Chapter 9 Review Exercises ....................................................................................... 1066 Chapter 9 Test ............................................................................................................. 1087

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    10 Analytic Geometry 10.1 Parabolas ............................................................................................................. 1095 10.2 Ellipses ................................................................................................................1107 Chapter 10 Quiz (Sections 10.1 − 10.2)........................................................................ 1118 10.3 Hyperbolas .......................................................................................................... 1120 10.4 Summary of the Conic Sections .......................................................................... 1134 Chapter 10 Review Exercises ..................................................................................... 1142 Chapter 10 Test ...........................................................................................................1152 11 Further Topics in Algebra 11.1 Sequences and Series .......................................................................................... 1156 11.2 Arithmetic Sequences and Series ........................................................................ 1170 11.3 Geometric Sequences and Series ........................................................................ 1179 Summary Exercises on Sequences and Series ............................................................ 1191 11.4 The Binomial Theorem ....................................................................................... 1193 11.5 Mathematical Induction ..................................................................................... 1200 Chapter 11 Quiz (Sections 11.1 − 11.5)........................................................................ 1214 11.6 Basics of Counting Theory .................................................................................1215 11.7 Basics of Probability ........................................................................................... 1223 Chapter 11 Review Exercises ..................................................................................... 1231 Chapter 11 Test ...........................................................................................................1243 Appendices Appendix A Polar Form of Conic Sections ................................................................1247 Appendix B Rotation of Axes .....................................................................................1250

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    1 Chapter R REVIEW OF BASIC CONCEPTS Section R.1 Sets 1. The elements of the set of natural number are {1, 2, 3, 4, …}. 2. Set A is a subset of set B if every element of set A is also an element of set B . 3. The set of all elements of the universal set U that do not belong to set A is the complement of set A . 4. The intersection of sets A and B is made up of all the elements belonging to both set A and set B . 5. The union of sets A and B is made up of all the elements belonging to set A or set B (or both). 6. The set   1 1 1 3 9 27 1, , , ,  is infinite. 7. Using set notation, the set { x | x is a natural number less than 6} is {1, 2, 3, 4, 5}. 8.   4, 5 A   9.       16,18, 21, 50 15,16,17,18 16,18   10.       16,18, 21, 50 15,16,17,18 15,16,17,18, 21, 50   11. The set {4, 5, 6, …, 15} has a limited number of elements, so it is a finite set. Yes, 10 is an element of the set. 12. The set {1, 2, 3, 4, 5, …, 75} has a limited number of elements, so it is a finite set. Yes, 10 is an element of the set. 13. The set   1 1 1 2 4 8 1, , , ,  has an unlimited number of elements, so it is an infinite set. No, 10 is not an element of the set. 14. The set {4, 5, 6, …} has an unlimited number of elements, so it is an infinite set. Yes, 10 is an element of the set. 15. The set { x | x is a natural number larger than 11}, which can also be written as {11, 12, 13, 14, …}, has an unlimited number of elements, so it is an infinite set. No, 10 is not an element of the set. 16. The set { x | x is a natural number greater than or equal to 10}, which can also be written as {10, 11, 12, 13, …}, has an unlimited number of elements, so it is an infinite set. Yes, 10 is an element of the set. 17. There are infinitely many fractions between 1 and 2, so { x | x is a fraction between 1 and 2} is an infinite set. No, 10 is not an element of the set. 18. The set { x | x is an even natural number} has no largest element. Because it has an unlimited number of elements, this is an infinite set. Yes, 10 is an element of the set. 19. The elements of the set {12, 13, 14, …, 20} are all the natural numbers from 12 to 20 inclusive. There are 9 elements in the set, {12, 13, 14, 15, 16, 17, 18, 19, 20}. 20. The elements of the set {8, 9, 10, …, 17} are all the natural numbers from 8 to 17 inclusive. There are 10 elements in the set, {8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. 21. Each element of the set   1 1 1 2 4 32 1, , , ,  after the first is found by multiplying the preceding number by 1 2 . There are 6 elements in the set,   1 1 1 1 1 2 4 8 16 32 1, , , , , . 22. Each element of the set {3, 9, 27, …, 729} after the first is found by multiplying the preceding number by 3. There are 6 elements in the set, {3, 9, 27, 81, 243, 729}. 23. To find the elements of the set {17, 22, 27, …, 47}, start with 17 and add 5 to find the next number. There are 7 elements in the set, {17, 22, 27, 32, 37, 42, 47}. 24. To find the elements of the set {74, 68, 62, …, 38}, start with 74 and subtract 6 (or add –6) to find the next number. There are 7 elements in the set, {74, 68, 62, 56, 50, 44, 38}. 25. When you list all elements in the set {all natural numbers greater than 8 and less than 15}, you obtain {9, 10, 11, 12, 13, 14}.

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    2 Chapter R Review of Basic Concepts 26. When you list all elements in the set {all natural numbers not greater than 4} you obtain {1, 2, 3, 4}. 27. 6 is an element of the set {3, 4, 5, 6}, so we write 6  {3, 4, 5, 6}. 28. 9 is an element of the set {2, 3, 5, 9, 8}, so we write 9  {2, 3, 5, 9, 8}. 29. 5 is not an element of {4, 6, 8, 10}, so we write 5  {4, 6, 8, 10}. 30. 13 is not an element of the set {3, 5, 12, 14}, so we write 13  {3, 5, 12, 14}. 31. 0 is an element of {0, 2, 3, 4}, so we write 0  {0, 2, 3, 4}. 32. 0 is an element of the set {0, 5, 6, 7, 8, 10}, so we write 0  {5, 6, 7, 8, 10}. 33. {3} is a subset of {2, 3, 4, 5}, not an element of {2, 3, 4, 5}, so we write {3}  {2, 3, 4, 5}. 34. {5} is a subset of {3, 4, 5, 6, 7}, not an element of {3, 4, 5, 6, 7}, so we write {5}  {3, 4, 5, 6, 7}. 35. {0} is a subset of {0, 1, 2, 5}, not an element of {0, 1, 2, 5}, so we write {0}  {0, 1, 2, 5}. 36. {2} is a subset of {2, 4, 6, 8}, not an element of {2, 4, 6, 8}, so we write {2}  {2, 4, 6, 8}. 37. 0 is not an element of ,  because the empty set contains no elements. Thus, 0 .  38.  is a subset of ,  not an element of .  The empty set contains no elements. Thus we write, .   39. False. 3 is not one of the elements in {2, 5, 6, 8}. 40. False. 6 is not one of the elements of {2, 5, 8, 9}. 41. True. 1 is one of the elements of {11, 5, 4, 3, 1}. 42. True. 12 is one of the elements of {18, 17, 15, 13, 12}. 43. True. 9 is not one of the elements of {8, 5, 2, 1}. 44. True. 3 is not an element of {7, 6, 5, 4}. 45. True. Both sets contain exactly the same four elements. 46. True. Both sets contain exactly the same five elements. 47. False. These two sets are not equal because {5, 8, 9, 0} contains the element 0, which is not an element of {5, 8, 9}. 48. False. These two sets are not equal because {3, 7, 12, 14, 0} contains the element 0, which is not an element of {3, 7, 12, 14}. 49. True. 1 and 2 are the only natural numbers less than 3. 50. True. Both sets describe the same elements. For Exercises 51 − 62, A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, and U = {2, 4, 6, 8, 10, 12, 14}. 51. True. This statement says “ A is a subset of U .” Because every element of A is also an element of U , the statement is true. 52. True. This statement says “ C is a subset of U .” Because every element of C is also an element of U , the statement is true. 53. True. Because both elements of D , 2 and 10, are also elements of B , D is a subset of B . 54. True. Because both elements of D , 2, and 10, are also elements of A , D is a subset of A . 55. False. Set A contains two elements, 6 and 12, that are not elements of B . Thus, A is not a subset of B . 56. False. Set B contains two elements, 2 and 8, that are not elements of C . Thus, B is not a subset of C . 57. True. The empty set is a subset of every set. 58. True. The empty set is a subset of every set. 59. True. Because 4, 8, and 10 are all elements of B , {4, 8, 10} is a subset of B . 60. False. Because 0 is not an element of D , {0, 2} is not a subset of D . 61. False. Because B contains two elements, 4 and 8, that are not elements of D , B is not a subset of D . 62. False. There are three elements of A (2, 6, and 8) that are not elements of C , so A is not a subset of C . 63. Every element of {2, 4, 6} is also an element of {2, 3, 4, 5, 6}, so {2, 4, 6} is a subset of {2, 3, 4, 5, 6}. We write {2, 4, 6}  {2, 3, 4, 5, 6}.

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    Section R.1 Sets 3 64. Every element of {1, 5} is also an element of {0, 1 2, 3, 5}, so {1, 5} is a subset of the set {0, 1 2, 3, 5}. We write {1, 5}  {0, 1 2, 3, 5}. 65. Because 0 is an element of {0, 1, 2}, but is not an element of {1, 2, 3, 4, 5}, {0, 1, 2} is not a subset of {1, 2, 3, 4, 5}. We write {0, 1, 2}  {1, 2, 3, 4, 5}. 66. Because 8 is an element of {5, 6, 7, 8}, but is not an element of {1, 2, 3, 4, 5, 6, 7}, {5, 6, 7, 8} is not a subset of {1, 2, 3, 4, 5, 6, 7}. We write {5, 6, 7, 8}  {1, 2, 3, 4, 5, 6, 7}. 67. The empty set is a subset of every set, so   {1, 4, 6, 8}. 68. The empty set is a subset of every set, including itself, so .    69. True. 7 and 9 are the only elements belonging to both sets. 70. True. 8 and 11 are the only elements belonging to both sets. 71. False. {1, 2, 7}  {1, 5, 9}  {1, 2, 5, 7, 9}, while {1, 2, 7}  {1, 5, 9}  {1}. 72. False. {6, 12, 14, 16}  {6, 14, 19}  {6, 12, 14, 16, 19}, while {6, 12, 14, 16}  {6, 14, 19}  {6, 14}. 73. True. 2 is the only element belonging to both sets. 74. False. The sets {6, 8, 9} and {9, 8, 6} are equal because they contain exactly the same three elements. Their union contains the same elements, namely 8, 9, and 6. 75. {3, 5, 9, 10}    {3, 5, 9, 10} In order to belong to the intersection of two sets, an element must belong to both sets. Because the empty set contains no elements, {3, 5, 9, 10} ,     so the statement is false. 76. True. For any set A, . A A    77. True. Because the two sets are equal, their union contains the same elements, namely 1, 2, and 4. 78. False. {1, 2, 4}  {1, 2, 4}  {1, 2, 4} 79. True. 80. True. For Exercises 81–110, U = {0, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}. 81. M  R The only elements belonging to both M and R are 0, 2, and 4, so M  R = {0, 2, 4}. 82. M  U Because M  U , the intersection of M and U will contain the same elements as M . M  U = M or {0, 2, 4, 6, 8}. 83. M  N The union of two sets contains all elements that belong to either set or to both sets. M  N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13} 84. M  R The union M and R is made up of elements which belong to M or to R (or to both). M  R = {0, 1, 2, 3, 4, 6, 8} 85. M  N There are no elements which belong to both M and N , so M  N .   M and N are disjoint sets. 86. U  N Because N  U, the elements belonging to U and N are all the elements belonging to N, U  N = N or {1, 3, 5, 7, 9, 11, 13} 87. N  R = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13} 88. M  Q Because M  Q , the elements belonging to M or Q are all the elements belonging to Q . M  Q = Q or {0, 2, 4, 6, 8, 10, 12} 89. N  The set N  is the complement of set N , which means the set of all elements in the universal set U that do not belong to N . N   Q or {0, 2, 4, 6, 8, 10, 12} 90. Q  The set Q  is the complement of set Q , which means the set of all elements in the universal set U that do not belong to Q . Q   N or {1, 3, 5, 7, 9, 11, 13}

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    4 Chapter R Review of Basic Concepts 91. M   Q First form M  , the complement of M . M  contains all elements of U that are not elements of M . Thus, M   {1, 3, 5, 7, 9, 10, 11, 12, 13}. Now form the intersection of M  and Q . Thus, we have M   Q  {10, 12}. 92. Q  R  First form R  , the complement of R . R  contains all elements of U the are not elements of R . Thus, R   {5, 6, 7, 8, 9, 10, 11, 12, 13}. Now form the intersection of Q and R  . Thus, we have Q  R   {6, 8, 10, 12} 93.   R Because the empty set contains no elements, there are no elements belonging to both  and R . Thus,  and R are disjoint sets, and . R     94.   Q Because the empty set contains no elements, there are no elements belonging to both  and Q . Thus,  and Q are disjoint sets, and . Q     95. N   Because  contains no elements, the only elements belonging to N or  are the elements of N . Thus,  and N are disjoint sets, and N  N   or {1, 3, 5, 7, 9, 11, 13}. 96. R   Because  contains no elements, the only elements belonging to R or  are the elements of R . Thus,  and R are disjoint sets, and R R    or {0, 1, 2, 3, 4}. 97. ( M  N )  R First, form the intersection of M and N . Because M and N have no common elements (they are disjoint), M  N .   Thus, ( M  N )  R . R    Now, because  contains no elements, the only elements belonging to R or  are the elements of R . Thus,  and R are disjoint sets, and R R    or {0, 1, 2, 3, 4}. 98. ( N  R )  M First form the union of N and R . We have N  R = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13}. Now form the intersection of this set with M . We have ( N  R )  M = {0, 2, 4}. 99. ( Q  M )  R First form the intersection of Q and M . We have Q  M = {0, 2, 4, 6, 8} = M. Now form the union of this set with R . We have ( Q  M )  R = M  R = {0, 1, 2, 3, 4, 6, 8}. 100. ( R  N )   M  First form the union of R and N . We have R  N = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13}. Now find the complement of M . We have M  = {1, 3, 5, 7, 9, 10, 11, 12, 13}. Now, find the intersection of these two sets. We have ( R  N ) M N    or {1, 3, 5, 7, 9, 11, 13}. 101. ( M   Q )  R First, find M  , the complement of M . We have M   {1, 3, 5, 7, 9, 10, 11, 12, 13}. Next, form the union of M  and Q . We have M   Q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} . U  Thus, we have ( M   Q ) R U R R     or {0, 1, 2, 3, 4}. 102. Q  ( M  N ) First, form the union of M and N . We have M  N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13}. Now form the intersection of Q with this set. We have Q  ( M  N ) M  or {0, 2, 4, 6, 8}. 103.   Q N U     First, find Q  , the complement of Q . We have Q   {1, 3, 5, 7, 9, 11, 13} . N  Now find N  , the complement of N . We have N   {0, 2, 4, 6, 8, 10, 12} . Q  Next, form the intersection of N  and U . We have N U Q U Q      Finally, we have   Q N U Q Q          Because the intersection of Q  and   N U   is ,  Q  and   N U   are disjoint sets. 104.   U R    Because , U    and , U U U   we have     U R U U R U R U           or {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. 105.   | , x x U x M   This means all elements of U except those in set M . This gives the set {1, 3, 5, 7, 9, 10, 11, 12, 13}.

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    Section R.2 Real Numbers and Their Properties 5 106.   | , x x U x R   This means all elements of U except those in set R . This gives the set {5, 6, 7, 8, 9, 10, 11, 12, 13}. 107.   | and x x M x Q   This means all elements that are members of both set M and set Q , or . M Q  Note that set M is a subset of set Q because all members of set M are included in set Q . Thus, the answer is set M or {0, 2, 4, 6, 8}. 108.   | and x x Q x R   This means all elements that are members of both set Q and set R , or . Q R  Thus, the answer is {0, 2, 4}. 109.   | or x x M x Q   This means all elements that are either members of set M or set Q , or . M Q  Note that set M is a subset of set Q because all members of set M are included in set Q . Thus, the answer is set Q or {0, 2, 4, 6, 8, 10, 12}. 110.   | or x x Q x R   This means all elements that are either members of set Q or set R , or . Q R  Thus, the answer is {0, 1, 2, 3, 4, 6, 8, 10, 12}. Section R.2 Real Numbers and Their Properties 1. {0, 1, 2, 3, …} describes the set of whole numbers. 2. {…, –3, –2, –1, 0, 1, 2, 3, …} describes the set of integers. 3. In the expression 3 6 , 6 is the base and 3 is the exponent. 4. If the real number a is to the left of the real number b on a number line, then a < (is less than) b . 5. The distance on a number line from a number to 0 is the absolute value of that number. 6. (a) 0 is a whole number. Therefore, it is also an integer, a rational number, and a real number. 0 belongs to B, C, D, F. (b) 34 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 34 belongs to A, B, C, D, F. (c) 9 4  is a rational number and a real number. 9 4  belongs to D, F. (d) 36 6  is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 36 belongs to A, B, C, D, F. (e) 13 is an irrational number and a real number. 13 belongs to E, F . (f) 216 54 100 25 2.16   is a rational number and a real number. 2.16 belongs to D, F. 7. 3 10 1000  8. 2 5 10 2 10 5 5       9. 4 4   10.   7 4 7 28 x y x y      For Exercises 11 − 16,   5 12 1 4 8 4 6, , , 3, 0, , 1, 2 , 3, 12 . A       11. 1 and 3 are natural numbers. 12. 0, 1, and 3 are whole numbers. 13. –6, 12 4  (or –3), 0, 1, and 3 are integers. 14. 5 12 1 4 8 4 6, (or 3), , 0, , 1, and 3     are rational numbers. 15. 3, 2 and 12   are irrational numbers. 16. All are real numbers. 17.   4 2 2 2 2 2 16         18.   5 3 3 3 3 3 3 243          19.           4 2 2 2 2 2 16           20.          6 2 2 2 2 2 2 2 64          21.             5 3 3 3 3 3 3 243              22. 5 ( 2) ( 2)( 2)( 2)( 2)( 2) 32          23.   4 2 3 2 3 3 3 3 2 81 162              24.   3 4 5 4 5 5 5 4 125 500            

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    6 Chapter R Review of Basic Concepts 25. 2 5 12 3 10 12 3 10 4 6              26. 9 3 16 4 27 16 4 27 4 23          27.                   3 3 4 9 8 7 2 4 1 7 2 4 1 7 8 4 7 8 4 56 60                         28.              4 6 5 3 2 6 5 3 16 30 3 16 30 48 30 48 18                      29. 3 (4 2 )( 2 25) (4 8)( 2 5) ( 4)(3) 12            30.             2 3 3 5 3 16 2 5 9 4 2 4 4 8 4 4 16             31. 2 1 5 1 9 4 18 2 8 9 5 9 36 36 18 18 17 4 17 8 25 36 18 36 36 36                                                               32. 5 2 3 11 8 5 2 10 25 16 15 11 9 4 40 40 10 10 40 10 9 16 25 5 40 40 40 8                                                               33.       8 4 6 12 8 24 12 4 3 4 3 8 2 7 6 6 7 7                    34.     15 5 4 6 8 3 4 6 8 6 5 8 2 6 5 4 12 6 8 2 8 6 5 4 1 4 6 6 5 5                                For Exercises 35 − 48, p = − 4, q = 8, and r = –10. 35.       2 2 2 2 2 7 4 7 8 10 4 7 8 100 16 7 8 100 16 56 100 72 100 28 p q r                              36.             2 2 2 4 2 8 10 16 2 8 10 16 16 10 32 10 42 p q r                            37. 8 ( 10) 2 1 8 ( 4) 4 2 q r q p            38. 4 ( 10) 14 7 4 8 4 2 p r p q             39. 3 5 3 8 5 24 5 12 5 10 4 10 4 5 4 12 5 48 25 23 5 4 20 20 20 q r p                       40. 3 2 3( 10) 2 30 2 8 10 8 10 15 1 75 4 71 4 5 20 20 20 r q r                        41. 5 5( 10) 50 2 3 2( 4) 3( 10) 8 30 50 25 22 11 r p r               42.         3 3 8 24 3 2 3 4 2 10 12 2 10 24 24 24 3 12 20 12 20 8 q p r                    43. 8 10 10 22 4 11 2 3 2 3 3 3 3 3( 4) 8 3 1 2 3 4 8 4 8 q r p q                44.   8 10 10 2 4 5 4 5 5 4 8 8 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 2 q r p q                     

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    Section R.2 Real Numbers and Their Properties 7 45.               2 2 2 2 3 4 2 3 10 2 2 8 2 3 10 6 4 3 10 4 30 6 6 4 30 26 13 6 6 3 p r q                                    46.           2 2 2 6 2 8 6 2 4 4 4 4 2 8 4 8 4 4 8 4 1 8 2 q p p                     47.             3 3 3 3 4 3 4 4 3 3 4 8 10 8 12 3 0 12 6 2 2 p p r                   48.             3 3 3 5 8 2 1 4 5 2 1 3 10 3 5 8 2 27 5 8 2 3 7 7 40 2 27 40 54 7 7 14 2 7 q p r                                     49. distributive 50. distributive 51. inverse 52. inverse 53. identity 54. identity 55. commutative 56. commutative 57. associative 58. associative 59. closure 60. closure 61. No; in general . a b b a    Examples will vary, i.e. if a = 15 and b = 0, then a − b = 15 − 0 = 15, but b − a = 0 − 15 = − 15. 62. No; in general     a b c a b c      . Examples will vary, i.e. if a = 15, b = 0, and c = 3, then     15 0 3 15 3 12, a b c         but       15 0 3 15 3 18. a b c          63.   10 10 22 22 20 11 11 z z z          64.     3 3 3 12 12 12 9 4 4 4 r r r r                           65.     5 6 5 6 11 m m m        66.       8 7 8 7 8 7 15 (or 15) a a a a a            67. 3 16 32 40 8 9 27 9 3 16 3 32 3 40 8 9 8 27 8 9 3 16 3 32 5 8 9 8 27 3 2 4 5 + 3 9 3 y z y z y z y z                                                 68.             1 20 8 32 4 1 1 1 20 8 32 4 4 4 1 1 1 20 8 32 4 4 4 5 + 2 8 = 5 2 8 m y z m y z m y z m y z m y z                                                         69. 8 14 (8 14) 6 p p p p      70.   15 10 15 10 5 x x x x     71.     4 4 4 4 4 z y z y z y          72.     3 3 3 3 3 m n m n m n          73. The process in your head should be the following:     72 17 28 17 (72 28) 17 (100) 17 1700        74. The process in your head should be the following:     32 80 32 20 32 80 20 32 100 3200        75. The process in your head should be the following: 5 1 5 1 5 5 1 123 1 23 1 123 23 1 8 2 8 2 8 8 2 1 (100) 1 150 2                         

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    8 Chapter R Review of Basic Concepts 76. The process in your head should be the following:   2 3 2 3 2 3 3 17 14 17 4 17 14 4 5 4 5 4 5 4 4 2 17 10 17.4 10 5 174                77. This statement is false because 6 8 2 2     and 6 8 6 8 2.      A corrected statement would be 6 8 6 8    or 6 8 8 6    78. This statement is false because   3 3 27 27     and 3 3 27 27      . A corrected statement would be   3 3 3 3    or   3 3 3 3 .   79. This statement is true because 5 6 5 6 30      and 5 6 30 30      . 80. This statement is true because 14 14 7 2 2    and 14 7 7 2     . 81. This statement is false. For example if you let 2 a  and 6 b  then 2 6 4 4     and 2 6 2 6 4 a b        . A corrected statement is a b b a    , if b > a > 0. 82. This statement is true by the algebraic definition of absolute value stated in the text. 83. 10 10   84. 15 15   85. 4 4 7 7    86. 7 7 2 2    87. 8 8     88. 12 12     89. 120 116 120 4 4 d P P        The d P value for a woman whose actual systolic blood pressure is 116 and whose normal value should be 120 is 4. 90. Consider the relation, 120 17. d P P    Because 103 and 137 both differ from 120 by 17, these are the two possible values for the patient’s systolic blood pressure. For Exercises 91 − 98, x = − 4 and y = 2. 91. 3 2 3( 4) 2(2) 12 4 16 16 x y           92.     2 5 2 4 5 2 8 10 18 18 x y           93. 3 4 3( 4) 4(2) 12 8 20 20 x y           94.       5 5 2 4 10 4 14 14 y x              95.       2 3 2 2 3 4 4 2 2 2 3 4 4 12 8 1 8 8 8 y x xy               96.     4 4 4 4 4 2 4 4 4 4 2 16 8 24 6 4 4 4 x y x            97. 8 8(2) ( 4) 16 ( 4) 4 4 20 20 5 4 4 y x x                       98.   2 4 2 2 4 2 2 8 2 4 4 4 x y x               99. True because 25 25  and 25 25.   100. True because the absolute value of any number is always greater than or equal to 0. 101. False.   5 13 8 8      while 5 13 5 13 18.      102. False. 8 12 4 4     while 8 12 8 12 4.      103. True because 11 6 11 6 66      and 66 66.   104. True because 10 5 5 2     and 10 10 5. 2 2   

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    Section R.2 Real Numbers and Their Properties 9 105.     , 1 4 1 4 3 3 d P Q          or     , 4 1 4 1 3 3 d P Q           106.     , 8 4 8 4 12 12 d P R        or   , 4 8 12 12 d P R       107.     , 8 1 8 1 9 9 d Q R        or   , 1 8 9 9 d Q R       108.     , 12 1 13 13 d Q S      or   , 1 12 13 13 d Q S       109. 0 xy  if x and y have the same sign. 110. 2 0 x y  if y is positive, because 2 x is positive for any nonzero x . 111. 0 x y  if x and y have different signs. 112. 2 0 y x  if x is negative, because 2 y is positive for any nonzero y . 113. Because 3 x has the same sign as x , 3 0 x y  if x and y have the same sign. 114. 0 x y   if x and y have different signs. 115. Because 18 ( 1) 19 19       and 1 ( 18) 19 19,     the number of strokes between their scores is 19. 116.   18, 355 3224 15 21, 564     yards. No, it is not the same, because the sum of the absolute values is 18, 355 3224 15 18, 355 3224 15 21, 594        The fact that | –15 | = 15 changes the two answers. 117. BAC 48 3.2 0.075 190 2 0.015 0.031        118. BAC 36 4.0 0.075 135 3 0.015 0.035        119. BAC 3 20 3.8 0.075 200 4 0.015 0.026         If the man’s weight was greater, the BAC would be lower. For example, if the man weighed 250 pounds, his BAC would be 3 20 3.8 0.075 250 4 0.015 0.0084.        120. BAC 2 6 14 0.075 150 2 0.015 0.054         If the woman consumed the same two glasses of wine over a longer period of time, her BAC would be lower. For example, if she drank the wine over 3 hours, her BAC would be 2 6 14 0.075 150 3 0.015 0.039.       

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    10 Chapter R Review of Basic Concepts For exercises 121 − 132, we use the formula 250 1000 12.5 6.25 1250 , 3 C T Y I A A A A R                                  where 0 0.775, 0 0.11875, 0 12.5, C T Y A A A       and 0 0.095. I A   121. First, find the value of the numerator. 250 1000 12.5 6.25 1250 304 34 3705 9 250 1000 12.5 6.25 1250 339.73 435 435 435 435 C T Y I A A A A                                                                   339.73 113.2 3 R   122. First, find the value of the numerator. 250 1000 12.5 6.25 1250 341 38 4381 5 250 1000 12.5 6.25 1250 336.56 520 520 520 520 C T Y I A A A A                                                                   336.56 112.2 3 R   123. First, find the value of the numerator. 250 1000 12.5 6.25 1250 408 32 4952 9 250 1000 12.5 6 6.25 1250 309.95 608 608 408 608 C T Y I A A A A                                                                   309.95 103.3 3 R   124. First, find the value of the numerator. 250 1000 12.5 6.25 1250 395 39 4727 15 250 1000 12.5 6.25 1250 304.55 597 597 597 597 C T Y I A A A A                                                                   304.55 101.5 3 R   125. First, find the value of the numerator. 250 1000 12.5 6.25 1250 373 33 4109 9 250 1000 12.5 6.25 1250 292.10 582 582 582 582 C T Y I A A A A                                                                   292.10 97.4 3 R  

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